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Scale 2065: "Motian"

Scale 2065: Motian, Ian Ring Music Theory

Bracelet Diagram

The bracelet shows tones that are in this scale, starting from the top (12 o'clock), going clockwise in ascending semitones. The "i" icon marks imperfect tones that do not have a tone a fifth above. Dotted lines indicate axes of symmetry.

Tonnetz Diagram

Tonnetz diagrams are popular in Neo-Riemannian theory. Notes are arranged in a lattice where perfect 5th intervals are from left to right, major third are northeast, and major 6th intervals are northwest. Other directions are inverse of their opposite. This diagram helps to visualize common triads (they're triangles) and circle-of-fifth relationships (horizontal lines).

Common Names




Cardinality is the count of how many pitches are in the scale.

3 (tritonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11


Forte Number

A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.


Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.


Reflection Axes

If a scale has an axis of reflective symmetry, then it can transform into itself by inversion. It also implies that the scale has Ridge Tones. Notably an axis of reflection can occur directly on a tone or half way between two tones.



A palindromic scale has the same pattern of intervals both ascending and descending.



A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

enantiomorph: 259


A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

1 (unhemitonic)


A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

0 (ancohemitonic)


An imperfection is a tone which does not have a perfect fifth above it in the scale. This value is the quantity of imperfections in this scale.



Modes are the rotational transformations of this scale. This number does not include the scale itself, so the number is usually one less than its cardinality; unless there are rotational symmetries then there are even fewer modes.


Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

prime: 35


Indicates if the scale can be constructed using a generator, and an origin.


Deep Scale

A deep scale is one where the interval vector has 6 different digits.


Interval Structure

Defines the scale as the sequence of intervals between one tone and the next.

[4, 7, 1]

Interval Vector

Describes the intervallic content of the scale, read from left to right as the number of occurences of each interval size from semitone, up to six semitones.

<1, 0, 0, 1, 1, 0>

Interval Spectrum

The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.


Distribution Spectra

Describes the specific interval sizes that exist for each generic interval size. Each generic <g> has a spectrum {n,...}. The Spectrum Width is the difference between the highest and lowest values in each spectrum.

<1> = {1,4,7}
<2> = {5,8,11}

Spectra Variation

Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.


Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.


Maximal Area Set

A scale is a maximal area set if a polygon described by vertices dodecimetrically placed around a circle produces the maximal interior area for scales of the same cardinality. All maximally even sets have maximal area, but not all maximal area sets are maximally even.


Interior Area

Area of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle, ie a circle with radius of 1.


Polygon Perimeter

Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.


Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.



A scale is balanced if the distribution of its tones would satisfy the "centrifuge problem", ie are placed such that it would balance on its centre point.


Ridge Tones

Ridge Tones are those that appear in all transpositions of a scale upon the members of that scale. Ridge Tones correspond directly with axes of reflective symmetry.



Also known as Rothenberg Propriety, named after its inventor. Propriety describes whether every specific interval is uniquely mapped to a generic interval. A scale is either "Proper", "Strictly Proper", or "Improper".


Heteromorphic Profile

Defined by Norman Carey (2002), the heteromorphic profile is an ordered triple of (c, a, d) where c is the number of contradictions, a is the number of ambiguities, and d is the number of differences. When c is zero, the scale is Proper. When a is also zero, the scale is Strictly Proper.

(1, 0, 6)

Common Triads

There are no common triads (major, minor, augmented and diminished) that can be formed using notes in this scale.


Modes are the rotational transformation of this scale. Scale 2065 can be rotated to make 2 other scales. The 1st mode is itself.

2nd mode:
Scale 385
Scale 385: Civian, Ian Ring Music TheoryCivian
3rd mode:
Scale 35
Scale 35: Abbian, Ian Ring Music TheoryAbbianThis is the prime mode


The prime form of this scale is Scale 35

Scale 35Scale 35: Abbian, Ian Ring Music TheoryAbbian


The tritonic modal family [2065, 385, 35] (Forte: 3-4) is the complement of the enneatonic modal family [959, 2023, 2527, 3059, 3311, 3577, 3703, 3899, 3997] (Forte: 9-4)


The inverse of a scale is a reflection using the root as its axis. The inverse of 2065 is 259

Scale 259Scale 259: Gijian, Ian Ring Music TheoryGijian


Only scales that are chiral will have an enantiomorph. Scale 2065 is chiral, and its enantiomorph is scale 259

Scale 259Scale 259: Gijian, Ian Ring Music TheoryGijian


In the abbreviation, the subscript number after "T" is the number of semitones of tranposition, "M" means the pitch class is multiplied by 5, and "I" means the result is inverted. Operation is an identical way to express the same thing; the syntax is <a,b> where each tone of the set x is transformed by the equation y = ax + b

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 2065       T0I <11,0> 259
T1 <1,1> 35      T1I <11,1> 518
T2 <1,2> 70      T2I <11,2> 1036
T3 <1,3> 140      T3I <11,3> 2072
T4 <1,4> 280      T4I <11,4> 49
T5 <1,5> 560      T5I <11,5> 98
T6 <1,6> 1120      T6I <11,6> 196
T7 <1,7> 2240      T7I <11,7> 392
T8 <1,8> 385      T8I <11,8> 784
T9 <1,9> 770      T9I <11,9> 1568
T10 <1,10> 1540      T10I <11,10> 3136
T11 <1,11> 3080      T11I <11,11> 2177
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 385      T0MI <7,0> 49
T1M <5,1> 770      T1MI <7,1> 98
T2M <5,2> 1540      T2MI <7,2> 196
T3M <5,3> 3080      T3MI <7,3> 392
T4M <5,4> 2065       T4MI <7,4> 784
T5M <5,5> 35      T5MI <7,5> 1568
T6M <5,6> 70      T6MI <7,6> 3136
T7M <5,7> 140      T7MI <7,7> 2177
T8M <5,8> 280      T8MI <7,8> 259
T9M <5,9> 560      T9MI <7,9> 518
T10M <5,10> 1120      T10MI <7,10> 1036
T11M <5,11> 2240      T11MI <7,11> 2072

The transformations that map this set to itself are: T0, T4M

Nearby Scales:

These are other scales that are similar to this one, created by adding a tone, removing a tone, or moving one note up or down a semitone.

Scale 2067Scale 2067: Movian, Ian Ring Music TheoryMovian
Scale 2069Scale 2069: Mowian, Ian Ring Music TheoryMowian
Scale 2073Scale 2073: Moyian, Ian Ring Music TheoryMoyian
Scale 2049Scale 2049: Major Seventh Ditone, Ian Ring Music TheoryMajor Seventh Ditone
Scale 2057Scale 2057: Mopian, Ian Ring Music TheoryMopian
Scale 2081Scale 2081: Modian, Ian Ring Music TheoryModian
Scale 2097Scale 2097: Munian, Ian Ring Music TheoryMunian
Scale 2129Scale 2129: Raga Nigamagamini, Ian Ring Music TheoryRaga Nigamagamini
Scale 2193Scale 2193: Major Seventh, Ian Ring Music TheoryMajor Seventh
Scale 2321Scale 2321: Zyphic, Ian Ring Music TheoryZyphic
Scale 2577Scale 2577: Punian, Ian Ring Music TheoryPunian
Scale 3089Scale 3089: Tirian, Ian Ring Music TheoryTirian
Scale 17Scale 17: Major Third Ditone, Ian Ring Music TheoryMajor Third Ditone
Scale 1041Scale 1041: Hitian, Ian Ring Music TheoryHitian

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow, graph visualization by Graphviz, and MIDI playback by MIDI.js. All other diagrams and visualizations are © Ian Ring. Some scale names used on this and other pages are ©2005 William Zeitler ( used with permission.

Pitch spelling algorithm employed here is adapted from a method by Uzay Bora, Baris Tekin Tezel, and Alper Vahaplar. (An algorithm for spelling the pitches of any musical scale) Contact authors Patent owner: Dokuz Eylül University, Used with Permission. Contact TTO

Tons of background resources contributed to the production of this summary; for a list of these peruse this Bibliography. Special thanks to Richard Repp for helping with technical accuracy, and George Howlett for assistance with the Carnatic ragas.